Let $T$ be the bounded linear operator on $l^1$ defined by $$T(x) = \Big ( \sum _{k \ge 2} x_k \Big) e_1 + \sum_{j=2}^\infty x_{j-1} e_j, $$
Show $\sigma_{ap}(T) \subseteq \bar{B}(0,1) \cup \{ \frac{1+\sqrt{5}}{2} \}$.
What I have: By solving $\lambda I - T = 0$, I could show $\sigma_{p}(T)= \{ \frac{1+\sqrt{5}}{2} \}$. Also, $||T|| \le 2$. But none of these give me what I need. Any hints?
Update 1:
I think it is possible to work with the dual operator. Noting that $\sigma(T)= \sigma_{ap}(T)\cup \sigma_p(T')$. It is easy to compute $\sigma_p(T')$ (comments below) in this case. Might this help?
Update 2: I have forgot to include the eigenvalue for $\sigma_{ap}(T)$.